Achilles number

An Achilles number is a powerful number greater than one that is not a perfect power. In its prime factorizationevery prime exponent satisfies $e_i\ge 2$, and $\gcd (e_1,\dots ,e_k)=1$. The first condition is precisely the definition of a powerful number. For the second, $n=m^r$ with integers $m,r>1$ exactly when $r$ divides every prime exponent; hence a greatest common divisor greater than one is precisely the obstruction.

For example, $72=2^3{3}^2$ is an Achilles number because $\gcd (3,2)=1$. By contrast, $144=2^4{3}^2=12^2$ is powerful but is a perfect square, so it is not an Achilles number.

A strong Achilles number is a positive integer $n$ for which both $n$ and its Euler totient $\phi (n)$ are Achilles numbers. For example,In each prime factorization, every prime exponent is at least two and their greatest common divisor is one. Thus both numbers are powerful without being perfect powers. The extra condition matters: $72$ is an Achilles number, but $\phi (72)=24=2^{3}3$ is not powerful, so $72$ is not a strong Achilles number.